Articulated Robots! Robert Stengel! Robotics and Intelligent Systems !MAE 345, Princeton University, 2013"• Robot Components"– Links and Joints"– Gearing and Leverage"• Robot Configurations"• Biological Antecedents"• Coordinate Systems"• Rotation Matrix"Copyright 2013 by Robert Stengel. All rights reserved. For educational use only.!http://www.princeton.edu/~stengel/MAE345.html!Human Anatomical Coordinates"Animal Anatomical Coordinates"Skeleton and Muscle-Induced Motion"Prosthetic Arms and Hands"Jesse with the DEKA/DARPA Arm, 2009!http://www.youtube.com/watch?v=ddInW6sm7JE!Jesse Sullivan, 2007!Rehabilitation Institute of Chicago!Robot Arms for Space"Canadarm, Space Shuttle!Dextre Manipulator, ISS!Canadarm2, ISS!Curiousity Robot Arm!http://www.youtube.com/watch?v=-kxF6Fo-EOg"Serial Robotic Manipulators"• Serial chain of robotic links and joints"– Large workspace"– Low stiffness"– Cumulative errors from link to link"– Proximal links carry the weight and load of distal links"– Actuation of proximal joints affects distal links"– Limited load-carrying capability at end effecter"– Proximal link: closer to the base!– Distal link: farther from the base!Baxter and the PR2"PR2!http://www.youtube.com/watch?v=HMx1xW2E4Gg"Baxter!http://www.youtube.com/watch?v=QHAMsalhIv8"Parallel Robotic Mechanisms"Stewart Platform!http://www.youtube.com/watch?v=QdKo9PYwGaU!• End plate is directly actuated by multiple links and joints (kinematic chains)"– Restricted workspace"– Common link-joint configuration"– Light construction"– Stiffness"– High load-carrying capacity"Pick-and-Place Robot!http://www.youtube.com/watch?v=i4oBExI2KiQ!Four-Bar Linkage"• Closed-loop structure"• Rotational joints"• Planar motion"• Proportions of link lengths determine pattern of motion"• Examples"– Double wishbone suspension"– Pantograph"– Scissor lift"See SimMechanics demo of four-bar linkage!Gearing and Leverage"Force multiplication"Displacement ratios"• Machine tools"– Grinding, sanding"– Inserting screws"– Drilling"– Hammering"• Paint sprayer"• Gripper, clamp"• Multi-digit hand"Prosthetic Hand"http://www.youtube.com/watch?v=QJg9igTnjIo&feature=related"End Effecters"Links and JointsRevolute Robotic Joints"Rotation about a single axis"Parallel to Link"Perpendicular to Link"Prismatic Robotic Joints"Sliding along a single axis"Other Robotic Joints"Cylindrical (sliding and turning composite)"Screw (helical motion)"Flexible"Spherical (or ball)"Planar (sliding and turning composite)"Universal"Constant-Velocity"Assembly Robot Configurations"McKerrow, 1991!Cartesian!Cylindrical!Polar! Articulated Revolute!Assembly Robot Workspaces"McKerrow, 1991!Tesla Model S Assembly"http://www.youtube.com/watch?v=8_lfxPI5ObM"Cylindrical!Articulated!Revolute!Characteristic Transformation of a Link"• Each link type has a characteristic transformation matrix relating the proximal joint to the distal joint"• Link n has"– Proximal end: Joint n, coordinate frame n – 1"– Distal end: Joint n + 1, coordinate frame n"Link: solid structure between two joints"McKerrow, 1991!Proximal!Distal!Links Between Revolute Joints - 1"• Link: solid structure between two joints"– Proximal end: closer to the base"– Distal end: farther from the base"• 4 Link Parameters"– Length of the link between rotational axes, l, along the common normal"– Twist angle between axes, α "– Angle between 2 links, θ (revolute)"– Offset between links, d (prismatic)"• Joint Variable: single link parameter that is free to vary"Type 1 Link "Two parallel revolute joints"Type 2 Link "Two non-parallel revolute joints"McKerrow, 1991!Links Between Revolute Joints - 2"Type 3 Link "Two revolute joints with intersecting rotational axes (e.g., shoulder)"Type 4 Link "Two perpendicular revolute joints with common origin (e.g., elbow-wrist)"• Link n extends along zn-1 axis"• ln = 0, along xn-1"• dn = length, along zn-1 (fixed)"• θn = variable joint angle n about zn-1!• αn = fixed orientation of n + 1 rotational axis about xn-1"McKerrow, 1991!• Link n extends along -zn axis"• ln = 0, along xn-1"• dn = 0, along zn-1"• θn = variable joint angle n about zn-1!• αn = fixed orientation of n + 1 rotational axis about xn-1"Links Involving Prismatic Joints - 1"Type 6 Link"Intersecting revolute "and prismatic joints"McKerrow, 1991!Type 5 Link "Intersecting prismatic joints"• Link n extends along zn-1 axis"• ln = 0, along xn-1"• dn = length, along zn-1 (variable)"• θn = 0, about zn-1!• αn = fixed orientation of n + 1 prismatic axis about xn-1"• Link n extends along zn-1 axis"• ln = 0, along xn-1"• dn = length, along zn-1 (fixed)"• θn = variable joint angle n about zn-1!• αn = fixed orientation of n + 1 prismatic axis about xn-1"Links Involving Prismatic Joints - 2"McKerrow, 1991!Type 7 Link "Parallel prismatic "and revolute joints"Type 8 Link"Intersecting prismatic "and revolute joints"• Link n extends along xn-1 axis"• ln = length along xn-1"• dn = 0, along zn-1"• θn = variable joint angle n about zn-1!• αn = 0, orientation of n + 1 prismatic axis about xn-1"• Link n extends along zn-1 axis"• ln = 0, along xn-1"• dn = length, along zn-1 (variable)"• θn = 0, about zn-1!• αn = fixed orientation of n + 1 rotational axis about xn-1"Robot Arm Reprise• Body reference planes• Animal coordinates• Components of a robot arm• Robotic mechanisms and configurations• Types of joints• Robot workspaces• Link-joint typesMath Review• Matrix and Transpose• Sums and Multiplication• Matrix Products• Identity Matrix• Matrix Inverse• TransformationsMatrix and Transpose"• Matrix: "– Usually bold capital or capital: F or F"– Dimension = (m x n)"• Transpose: "– Interchange rows and columns"A =a b cd e fg h kl m n⎡⎣⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥AT=a d g lb e h mc f k n⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥4 x 3!3 x 4!y = Ax =a b cd e fg h kl m n⎡⎣⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥x1x2x3⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥=y1y2y3y4⎡⎣⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥Matrix